The integral zero points are related to such a loop, but the \( q \BA \) portion of the momentum \( \Bp – q \BA \) needs to be considered.

Superconductors

After cooling some materials sufficiently, superconductivity, a complete lack of resistance to electrical flow can be observed. A resistivity vs temperature plot of such a material is sketched in fig. 4.

fig. 4. Superconductivity with comparison to superfluidity

Just like \ce{He^4} can undergo Bose condensation, superconductivity can be explained by a hybrid Bosonic state where electrons are paired into one state containing integral spin.

The Little-Parks experiment puts a superconducting ring around a magnetic whisker as sketched in fig. 6.

fig. 6. Little-Parks superconducting ring

This experiment shows that the effective charge of the circulating charge was \( 2 e \), validating the concept of Cooper-pairing, the Bosonic combination (integral spin) of electrons in superconduction.

But what is this \( x_0 \)? Because \( k_y \) is not really specified in this problem, we can consider that we have a zero point energy for every \( k_y \), but the oscillator position is shifted for every such value of \( k_y \). For each set of energy levels fig. 8 we can consider that there is a different zero point energy for each possible \( k_y \).

fig. 8. Energy levels, and Energy vs flux

This is an infinitely degenerate system with an infinite number of states for any given energy level.

This tells us that there is a problem, and have to reconsider the assumption that any \( k_y \) is acceptable.

To resolve this we can introduce periodic boundary conditions, imagining that a square is rotated in space forming a cylinder as sketched in fig. 9.